The Free-Electron MO Method

The simplest semiempirical w-electron theory is the free-electron molecular-orbital (FE MO) method, developed about 1950. Here the interelectronic repulsions l/r,y are ignored, and the effect of the cr electrons is represented by a particle-in-a-box potential-energy function V = 0 in a certain region, while V = oo outside this region. With the interelectronic repulsions omitted, in (16.1) becomes the sum of Hamiltonians for each electron hence (Section 6.2) 

The wave function (16.4) takes no account of spin or the Pauli principle. To do so, we must put each electron in a spin-orbital m, = where cr, is a spin function (either a or j8). The wave function is then written as an antisymmetrized product (Slater determinant) of spin-orbitals. Since does not involve spin, we have 

For conjugated chain molecules, the FE MO approximation takes the box in which the tt electrons move as one dimensional. We have [Eq. (2.20)] 

s are restricted by the Pauli principle no more than two electrons can be in a given spatial MO. In the ground electronic state, the tt electrons fill the n lowest free-electron TT MOs. 

The TT-electron transition giving the lowest-frequency electronic absorption involves an electron going from the highest occupied to the lowest vacant MO. For example, in a ground-state molecule with six tt electrons, the tt electrons fill the lowest three FE MOs, and the lowest-frequency transition has one electron going from the = 3 to the n = 4 FE MO. The particle-in-a-box selection rule (Section 9.10) is that An 

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The perturbational MO method of Longuet-Higgins (11) and Dewar (12), which was thoroughly reviewed by Dewar and Dougherty (6), has been the pencil-and-paper method of choice in numerous applications. More recently, a modified free-electron (MFE) MO approach (13-15) and a valence-bond structure-resonance theory (VBSRT) (7, 16, 17) have been applied to several PAH structure and reactivity problems. A new perturbational variant of the free-electron MO method (PMO F) has also been derived and reported (8, 18). Both PMO F and VBSRT qualify as simple pencil-and-paper procedures. When applied to a compilation of electrophilic substitution parameters (ct+) (19-23), the correlation coefficients of calculated reactivity indexes with cr+ for alternant hydrocarbons are 0.973 and 0.959, respectively (8). In this case, the performance of the PMO F method rivals that of the best available SCF calculations for systems of this size, and that of VBSRT is sufficient for most purposes. 

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